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Saturday, August 1, 2026 · Weekly Paper, Week 6

Still the Referee: IBM's Quantum Chemistry and a Number Nobody Can Use

IBM is spending heavily on quantum computing, and spending is easy to mistake for an advantage. This week I went past the roadmap and into the science: a July 2026 PNAS paper from IBM Research and the University of Chicago introducing LASSQD, a hybrid workflow that keeps a classical orbital-optimization loop and replaces only the fragment solver, letting a quantum processor supply samples that a classical computer turns into a small subspace to diagonalize. The engineering works, an instability is found and honestly fixed, and hardware keeps pace with noiseless simulation. Then the one calculation beyond exact classical fragment diagonalization returns a spin gap of 85 plus or minus 41 kcal/mol, in a paper that treats 1 kcal/mol as its accuracy reference, without ever computing the triplet state that theory and experiment favor. What follows is what the processor actually does, what connectivity costs, what the four validation layers show, and where that leaves the investment case. This is a first reading of one paper, written to learn in public. It is my own opinion, and it is not a valuation and not financial advice. Do your own research.

Investment is not an edge

IBM is spending heavily on quantum computing. That is a fact, and it is easy to mistake for an advantage. Spending buys a roadmap. It does not buy a technological lead, and the two get quoted as if they were the same thing.

So this week I went past the roadmap and into the science. The paper is Wang and colleagues, published in the Proceedings of the National Academy of Sciences in July 2026, written by IBM Research together with the University of Chicago: "Localized sample-based quantum diagonalization for strongly correlated chemistry."1 My question was narrow. When a quantum computer is pointed at a real application, what does IBM's technology actually contribute?

Why this chemistry is hard

Many molecules can be described reasonably well by a mean-field picture, in which each electron moves in the average field of the others. Strongly correlated transition-metal complexes can break that approximation, because no single electron configuration is enough. Many configurations contribute at once, and the true state is a weighted combination of them.

To calculate such a state, researchers choose an active space: a finite set of orbitals whose electron configurations are treated explicitly, while the remaining orbitals are treated more approximately. Each distinct way of distributing the electrons among those orbitals is one configuration. The matrix holding the energies and interactions among all of them is the HamiltonianThe operator describing a system's total energy and how it changes over time. Writing one down is usually the first step in stating a quantum problem., and diagonalizing it, which means finding the particular combinations of configurations that behave as stable states, gives those states and their energies. The lowest is the ground state. Doing this exactly is called full configuration interaction, or FCI, and it is exact only within the chosen finite orbital basis and active space.

The problem is counting. For $M$ orbitals holding $N_\alpha$ spin-up and $N_\beta$ spin-down electrons, the number of configurations is

$$D \;=\; \binom{M}{N_\alpha}\binom{M}{N_\beta}.$$

Those are ordinary binomial coefficients, the number of ways to choose which orbitals are filled. For the singlet porphyrin fragment, 26 electrons in 24 spatial orbitals, that comes to about $6.2 \times 10^{12}$ determinants.2 You cannot build that matrix, let alone diagonalize it. That single line of arithmetic is why a quantum computer is in this story at all.

The quantum processor never solves the molecule

Here is what surprised me most. The chip does not diagonalize the HamiltonianThe operator describing a system's total energy and how it changes over time. Writing one down is usually the first step in stating a quantum problem.. It does not even try.

The classical baseline is a method called LASSCF, localized active space self-consistent field. It partitions the active space into fragments, solves each fragment with FCI, updates the orbitals, and repeats until the answer stops moving. The hybrid method, LASSQD, keeps that orbital-optimization structure and replaces the fragment solver: instead of solving a fragment exactly, it uses sample-based quantum diagonalization, or SQD.

The chip never solves the molecule. It helps tell the classical computer where to look.

SQD works like this. A quantum circuitA sequence of quantum gates applied to qubits, read left to right in time and ending in measurement. The recipe a quantum computer runs. prepares a correlated electronic state and is measured many times, each measurementA physical process that produces a classical outcome and updates the quantum state. In an ideal projective measurement, the state is left in an eigenstate associated with the observed outcome. returning one bitstringThe string of 0s and 1s returned by a computational-basis readout. It is raw measurement data that must be interpreted according to the operator and measurement basis.. Those samples define a subset $\mathcal{S}$ of the enormous configuration space, and a classical computer then builds and diagonalizes only

$$\hat{H}_{\mathcal{S}} \;=\; \hat{P}_{\mathcal{S}}\,\hat{H}\,\hat{P}_{\mathcal{S}}, \qquad \hat{P}_{\mathcal{S}} \;=\; \sum_{x \in \mathcal{S}} |x\rangle\langle x| .$$

The operator $\hat{P}_{\mathcal{S}}$ is a projector: it keeps the configurations in $\mathcal{S}$ and discards everything else. It matters who builds $\mathcal{S}$. The quantum processor supplies samples biased toward configurations that may matter. Classical SQD processing then uses those samples to recover valid configurations, construct the selected subspace, diagonalize the Hamiltonian within it, and return the density matrices needed for the next orbital update.

So the hoped-for edge is not that quantum replaces classical. It is that quantum helps tell classical where not to look. In this paper the processor is the sidekick, not the lead.

LASSCF: classical baseline Fixed fragment partition Solve fragment: FCI Update orbitals LASSQD: hybrid workflow Fixed fragment partition Solve fragment: SQD Update orbitals fragment solver changes Quantum: prepare + measure LUCJ state Classical: process samples + diagonalize selected subspace LASSCF: classical baseline Fixed fragment partition Solve fragment: FCI Update orbitals LASSQD: hybrid workflow Fixed fragment partition Solve fragment: SQD Quantum: prepare + measureLUCJ state Classical: process samples +diagonalize selected subspace Update orbitals fragment solver changes
Figure 1: The same orbital-optimization structure with a different fragment solver. LASSQD replaces fragment FCI with quantum-assisted SQD while retaining the classical orbital-update loop.

What agreement with LASSCF means

Agreement with LASSCF needs an important qualification, and it is the thing I nearly missed. LASSQD is reproducing the fragmented LASSCF model, not exact molecular chemistry. Both methods solve strong correlation inside each fragment, but they treat correlation between fragments only at the mean-field level. In the paper's small benchmark, an unfragmented CASCI calculation using the same LASSCF-optimized orbitals lowered the energy by about 0.156 HartreeThe natural unit of energy in atomic physics and chemistry. One Hartree is about 627 kcal/mol, so chemically important energy differences are a tiny fraction of it., close to 98 kcal/mol. In that benchmark, the interfragment correlation omitted by LASSCF was therefore much larger than the numerical solver errors discussed later.

That does not invalidate the workflow. It defines what "agreement with the classical answer" actually means.

Connectivity, not qubit count

The circuit that prepares the state is called LUCJ, for local unitary cluster Jastrow. Its interactions have to be laid onto real hardware, and the hardware has opinions. On IBM's heavy-hex lattice each physical qubitAn actual piece of qubit hardware. On its own it is fragile and makes frequent errors. has at most three direct neighbors. When the pattern a circuit wants does not match the pattern the chip provides, the compiler inserts routing operations to shuttle information around. That adds circuit depthThe number of sequential layers of gates in a quantum circuit. Greater depth generally means a longer computation and more exposure to noise., and depth adds noise. This is why LUCJ restricts many of its interactions to nearby orbitals. It is shaped to fit the chip.

The hardware has opinions. The circuit is shaped to fit them.

The qubitThe basic unit of a quantum computer. Like a 'bit' in a normal computer, but instead of being only 0 or 1 it can be 0, 1, or a blend of both at once. maps made that concrete for me. The 127-qubit Sherbrooke processor runs a circuit that uses a smaller set of qubits, spread across the device with unused qubits between them. Some encode whether a spin orbital is occupied; others are auxiliary. The paper compares no alternative placements, so this is not evidence that the layout is poor. It is evidence of something narrower: usable problem size depends not only on how many physical qubits exist, but also on whether the circuit's required interactions can be mapped efficiently onto the available connectivityWhich qubits in a machine can directly interact with each other. More connectivity makes more algorithms possible.. Gate fidelityA score (for example 99.9%) for how accurately a quantum gate does what it is supposed to do. Higher means fewer errors., circuit depth, sampling cost, calibrationTuning the control signals so gates stay accurate. It drifts over time and must be redone. and classical postprocessing all bind as well.

Qubit countThe raw number of physical qubits in a machine. A weak measure of power on its own, since quality and connectivity matter more. reaches the headline. Connectivity reaches the calculation.

at most three direct neighbors example circuit qubits auxiliary unused in this example at most three direct neighbors example circuit qubits auxiliary unused in this example
Figure 2: Schematic heavy-hex connectivity, not the actual Sherbrooke layout. Usable circuit size depends not only on physical-qubit count but also on whether the required interactions can be mapped efficiently onto the available couplings.

Four layers, and one honest fix

The paper does not report a single result. It tests four things, and that structure is the most impressive part of the work.

First, whether the hybrid loop lands where the classical one does. On the smallest fragments, LASSQD matched LASSCF to within about $5 \times 10^{-7}$ HartreeThe natural unit of energy in atomic physics and chemistry. One Hartree is about 627 kcal/mol, so chemically important energy differences are a tiny fraction of it.. That test comes with a qualification the paper is clear about: the sampled bitstringsThe string of 0s and 1s returned by a computational-basis readout. It is raw measurement data that must be interpreted according to the operator and measurement basis. covered the full fragment configuration space, so SQD was formally equivalent to the exact fragment solver there. It is a validation result, not evidence that sparse quantum selection reduced the problem. Better still, four separate hybrid runs with different settings all produced orbitals that, re-solved classically, reproduced the classical energy to seven decimal places. The orbital optimization survives an approximate solver.

Second, whether the method is stable. It was not. Because each cycle drew a fresh sample, the energy oscillated from cycle to cycle, and after discarding the first twenty cycles the average error still exceeded 3 kcal/mol. The authors state plainly that conventional SQD is unsuitable for stable orbital optimization. Their fix, called carryover, retains the important configurations from one cycle into the next, after which the error falls below 1 kcal/mol within roughly nine cycles and converges smoothly. Reporting a failure and then fixing it is good science, and I want to say so before I get critical.

Third and fourth, whether this survives on real hardware and whether quantum-selected subspaces compete with classically selected ones. On IBM's Torino processor, carryover LASSQD reached the sub-1 kcal/mol agreement target after nine macrocycles, broadly consistent with the noiseless simulations. Against a classical selected-configuration method, the quantum-sampled subspace reached comparable accuracy, and lower energies in a middle band of subspace sizes, though that comparison is made at comparable selected-subspace dimensions, not at equal runtime or total solver effort.

The price

Then comes iron-porphyrin, the calculation the whole paper builds toward, where exact classical fragment diagonalization is out of reach.

Be careful about what that means. Exact classical fragment diagonalization could not be run. The authors instead used 23 LASSCI calculations, 12 for the quintet and 11 for the singlet. Quantum resource constraints limited LASSQD hardware to five calculations per spin state, 10 total. The classical extrapolation came back as 96 kcal/mol with an uncertainty of 11, the hybrid one as 85 with an uncertainty of 41.

Those plus-or-minus values are energy-variance extrapolation uncertainties. They describe how well each extrapolation was constrained. They do not include the missing triplet, dynamic correlation outside the active space, the mean-field treatment of interfragment correlation, or the choice of active space and fragmentation. The quoted ±40.7 kcal/mol is roughly forty times the 1 kcal/mol accuracy reference the paper uses earlier.

The central estimate favors the quintet over the singlet, but the quoted uncertainty is about half the value. More importantly, the triplet state favored by density functional theory and some experiments on porphyrin derivatives was not included. The calculation therefore cannot determine the actual ground state.

To their credit the authors say most of this. They describe the work as a feasibility demonstration and write that the error bars do not yet support predictive superiority.

One more thing about cost, because it is routinely misread. The paper demonstrates that SQD can reduce the dimension of the HamiltonianThe operator describing a system's total energy and how it changes over time. Writing one down is usually the first step in stating a quantum problem. that must be diagonalized. It does not demonstrate a faster or cheaper end-to-end hybrid calculation, and a smaller diagonalization subspace does not by itself mean lower total cost once circuit execution, sampling, configuration recovery, repeated macrocycles and classical postprocessing are counted.

A Extrapolated singlet-quintet gap LASSCI 96 ± 11 LASSQD hardware 85 ± 41 050100150 kcal/mol B Quoted uncertainty versus earlier accuracy reference LASSCI 10.7 LASSQD hardware 40.7 1 kcal/mol benchmark accuracy reference used earlier in the paper 01020304045 kcal/mol A Extrapolated singlet-quintet gap LASSCI 96 ± 11 LASSQD hardware 85 ± 41 050100150 kcal/mol B Quoted uncertainty versus earlier accuracy reference LASSCI 10.7 LASSQD hardware 40.7 1 kcal/mol benchmark accuracy reference used earlier in the paper 01020304045 kcal/mol
Figure 3: Panel A compares the extrapolated singlet-quintet gaps. Panel B compares the quoted extrapolation uncertainties with the paper’s earlier 1 kcal/mol benchmark accuracy reference. The LASSQD hardware uncertainty is about four times the LASSCI uncertainty and about forty times the earlier 1 kcal/mol reference.

My takeaway. The engineering worked and the instability fix is real. But the paper's main accuracy claims are anchored to classical references, and in the one regime where the exact classical solver could not follow, the result cannot identify the ground state. If the goal was to show the calculation can be run, it succeeded. If the goal was to show it should be run, nothing here does that yet.

The investor's read

IBM's own position is not ambiguous. In its second-quarter 2026 earnings remarks the company said quantum computing is no longer decades away and that it is investing aggressively, disclosing plans to invest more than $10 billion in quantum over the next five years across research, capital expenditure, manufacturing, acquisitions and ecosystem.3 It announced a letter of intent with the US Department of Commerce to build a dedicated quantum wafer foundryA factory that manufactures chips. Owning one lets a company iterate its own hardware faster than outsourcing., supported by $1 billion in CHIPS incentives and a matching $1 billion of IBM cash, and restated its target of delivering a large-scale fault-tolerant quantum computer, Starling, by 2029.4 It also cited an IDC evaluation of eleven quantum vendors that ranked IBM first overall. These are the company's own claims and IDC's assessment, not independently verified results.

Nine years on, the classical method is still the referee.

Set against that, here is what the financial disclosure contains. IBM's disclosed operating economics remain in Software, Consulting, Infrastructure and Financing. Quantum is not separately reported with its own revenueThe total money a company brings in from sales, before any costs are subtracted., margin or backlog. Software is nearly 45% of IBM's revenue, second-quarter revenue grew 1%, and the company guided to four to five percent growth for the full year.3

Proof-of-concept quantum chemistry on hardware is also not new. IBM reported variational molecular-energy calculations through beryllium hydride in 2017.5 Nam and colleagues published a trapped-ion water-molecule calculation in 2020.6 The scale and sophistication have advanced substantially. The validation pattern remains familiar: run the quantum-assisted calculation, compare it with a classical reference and report agreement. Nine years on, the classical method is still the referee.

Two days before I published this, IBM announced three partner quantum-advantage preprints.7 I have not evaluated them here. They may be more relevant to IBM's current advantage case than this chemistry paper is, and I will treat them separately rather than judge them from an announcement. One of those results has been exposed publicly through IBM's Quantum AdvantageA quantum computer solving a genuinely useful problem faster, cheaper, or better than the best classical computer. Tracker for months, and by IBM's account no classical method has yet reproduced it across the full regime. That kind of public pressure-testing is exactly how such claims should be evaluated, and I take the invitation seriously without treating the absence of a refutation as independent validation.

Where I land

As a physicist I find the construction elegant. Partition a molecule, let a quantum processor help propose which corners of an unimaginably large space are worth examining, and hand a tractable matrix back to a classical computer. Building two classical controls specifically to test whether the quantum selection was doing anything is more honest than much of what gets published in this field.

As an investor I come out where I started, only more firmly. This is a proof of concept, and it was not what I expected from one of the industry's largest and longest-running quantum programs. A single paper is not a verdict, and I would be glad to be shown what I am not seeing. But it does not move IBM's quantum investment case forward for me.

I still value IBM on its core business: the software, the consulting, the mainframes, the things that pay today. The quantum program is real, seriously funded, and publishing serious science, and none of that is yet a revenueThe total money a company brings in from sales, before any costs are subtracted. engine I would put a number on. I need to see more from the quantum team before I treat it as one.

Sources & notes

  1. Q. Wang, K. J. Sung, R. D'Cunha, M. R. Hermes, T. Gujarati, Y. Kawashima, Y. Ohnishi, G. O. Jones, M. Motta and L. Gagliardi, "Localized sample-based quantum diagonalization for strongly correlated chemistry," Proceedings of the National Academy of Sciences 123(28), e2603914123 (2026). https://doi.org/10.1073/pnas.2603914123
  2. The paper is internally inconsistent about this fragment. The main text, Fig. 6 and the principal results use a (26e,24o) porphyrin-ring fragment, while Table 4 lists (26e,23o). This article follows the main results and uses 24 orbitals.
  3. IBM, "2Q26 Earnings Prepared Remarks," July 22, 2026, pp. 2, 5 and 6. Company statements on quantum investment, the Starling roadmap and the IDC ranking are IBM's own claims and are not independently verified.
  4. A. Krishna, "Letter to IBM Investors," IBM Newsroom, July 14, 2026. https://newsroom.ibm.com/2026-07-14-Arvind-Krishnas-Letter-to-IBM-Investors
  5. A. Kandala, A. Mezzacapo, K. Temme, M. Takita, M. Brink, J. M. Chow and J. M. Gambetta, "Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets," Nature 549, 242-246 (2017). https://doi.org/10.1038/nature23879
  6. Y. Nam, J.-S. Chen, N. C. Pisenti et al., "Ground-state energy estimation of the water molecule on a trapped-ion quantum computer," npj Quantum Information 6, 33 (2020). https://doi.org/10.1038/s41534-020-0259-3
  7. IBM Newsroom, July 30, 2026, announcing quantum-advantage demonstrations with Qedma (arXiv:2607.24937), Algorithmiq (arXiv:2607.25998) and the University of Chicago, "Sampling hard circuits with verifiably high fidelity" (arXiv:2607.25941). These are preprints and company announcements, not peer-reviewed results.